Theorem euor2 1414

Description: Introduce or eliminate a disjunct in a uniqueness quantifier.

Assertion

Ref	Expression
euor2	|- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		euex 1371	. . . . . . 7 |- (E!x(ph \/ ps) -> E.x(ph \/ ps))
2		19.43 1064	. . . . . . 7 |- (E.x(ph \/ ps) <-> (E.xph \/ E.xps))
3	1, 2	sylib 198	. . . . . 6 |- (E!x(ph \/ ps) -> (E.xph \/ E.xps))
4	3	ord 232	. . . . 5 |- (E!x(ph \/ ps) -> (-. E.xph -> E.xps))
5	4	com12 11	. . . 4 |- (-. E.xph -> (E!x(ph \/ ps) -> E.xps))
6		eumo 1388	. . . . . 6 |- (E!x(ph \/ ps) -> E*x(ph \/ ps))
7		orcom 246	. . . . . . . 8 |- ((ph \/ ps) <-> (ps \/ ph))
8	7	mobii 1382	. . . . . . 7 |- (E*x(ph \/ ps) <-> E*x(ps \/ ph))
9		moor 1401	. . . . . . 7 |- (E*x(ps \/ ph) -> E*xps)
10	8, 9	sylbi 199	. . . . . 6 |- (E*x(ph \/ ps) -> E*xps)
11	6, 10	syl 10	. . . . 5 |- (E!x(ph \/ ps) -> E*xps)
12	11	a1i 8	. . . 4 |- (-. E.xph -> (E!x(ph \/ ps) -> E*xps))
13	5, 12	jcad 598	. . 3 |- (-. E.xph -> (E!x(ph \/ ps) -> (E.xps /\ E*xps)))
14		eu5 1386	. . 3 |- (E!xps <-> (E.xps /\ E*xps))
15	13, 14	syl6ibr 213	. 2 |- (-. E.xph -> (E!x(ph \/ ps) -> E!xps))
16		hbe1 990	. . . . 5 |- (E.xph -> A.xE.xph)
17	16	euor 1375	. . . 4 |- ((-. E.xph /\ E!xps) -> E!x(E.xph \/ ps))
18		euex 1371	. . . . . 6 |- (E!xps -> E.xps)
19		olc 268	. . . . . . 7 |- (ps -> (ph \/ ps))
20	19	19.22i 1016	. . . . . 6 |- (E.xps -> E.x(ph \/ ps))
21		19.8a 1005	. . . . . . . . 9 |- (ph -> E.xph)
22	21	orim1i 337	. . . . . . . 8 |- ((ph \/ ps) -> (E.xph \/ ps))
23	22	ax-gen 955	. . . . . . 7 |- A.x((ph \/ ps) -> (E.xph \/ ps))
24		euim 1398	. . . . . . 7 |- ((E.x(ph \/ ps) /\ A.x((ph \/ ps) -> (E.xph \/ ps))) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
25	23, 24	mpan2 693	. . . . . 6 |- (E.x(ph \/ ps) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
26	18, 20, 25	3syl 20	. . . . 5 |- (E!xps -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
27	26	adantl 388	. . . 4 |- ((-. E.xph /\ E!xps) -> (E!x(E.xph \/ ps) -> E!x(ph \/ ps)))
28	17, 27	mpd 26	. . 3 |- ((-. E.xph /\ E!xps) -> E!x(ph \/ ps))
29	28	ex 373	. 2 |- (-. E.xph -> (E!xps -> E!x(ph \/ ps)))
30	15, 29	impbid 514	1 |- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 950  E.wex 956  E!weu 1357  E*wmo 1358

This theorem is referenced by:  reuun2 2249

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360

Copyright terms: Public domain